A) \[\frac{{{\omega }^{2}}{{R}^{2}}{{d}^{2}}}{2}\]
B) \[\frac{\omega R{{d}^{2}}}{2}\]
C) \[\frac{{{\omega }^{2}}Rd}{2}\]
D) \[\frac{{{\omega }^{2}}{{R}^{2}}d}{2}\]
Correct Answer: D
Solution :
When an incompressible and non-viscous liquids flows in stream lined motion from one place to another then at every point of its path the total energy per unit volume is constant. \[P+\frac{1}{2}\rho {{v}^{2}}+\rho gh=\text{constant}\] where P is pressure, \[\rho \]is density, v is velocity and h is height. Also \[v=R\omega \] where R is radius and o is angular velocity. Since, velocity at centre is zero and density \[\rho =d,\]we have \[{{P}_{1}}+\frac{1}{2}\times 0={{P}_{2}}+\frac{1}{2}d{{v}^{2}}\] \[{{P}_{1}}={{P}_{2}}+\frac{1}{2}d{{(R\omega )}^{2}}\] Thus, increase in pressure \[{{P}_{1}}-{{P}_{2}}=\frac{1}{2}d{{R}^{2}}{{\omega }^{2}}\]
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